Find the mean of the distribution shown.
x 2 5 7
P(x) 0.28 0.42 0.30
What is the mean and the standard deviation of this distribution?
To find the mean of a distribution, you need to multiply each value by its corresponding probability, then sum these products.
In this case, we'll use the formula for mean (µ):
µ = ∑(x * P(x))
Where x represents each value, and P(x) represents the probability of each value.
So let's calculate it:
(2 * 0.28) + (5 * 0.42) + (7 * 0.30) = 0.56 + 2.10 + 2.10 = 4.76
Therefore, the mean of the distribution is 4.76.
Now, to find the standard deviation, you need to calculate the variance first. The formula for variance (σ²) is:
σ² = ∑((x - µ)² * P(x))
Where x represents each value, µ is the mean, and P(x) is the probability of each value.
First, let's calculate the variance:
((2 - 4.76)² * 0.28) + ((5 - 4.76)² * 0.42) + ((7 - 4.76)² * 0.30)
= (2.76² * 0.28) + (0.24² * 0.42) + (2.24² * 0.30)
= 2.4376 + 0.0108 + 1.9352
= 4.3836
Then, the standard deviation (σ) is the square root of the variance:
σ = √(σ²) = √(4.3836) = 2.09 (approx.)
Therefore, the standard deviation of the distribution is approximately 2.09.